How to perfectly align UV coordinates automatically?
Follow Active Quads is indeed the best automatic preset to use in this situation, however its use is slightly nuanced, and until recently I didn't know how to properly use it myself.

Start by erasing your existing UV Map from the Properties Window > Object Data > UV Maps, so you can start afresh.

Select one random face, any four sided face will do, and unwrap it using the basic U > Unwrap > Reset option so it is unwraped as a perfect square.

This will also make it the nominal active quad, only then can the Follow Active Quads option be used since it will use the active face's coordinates to unwrap the remaining mesh.

Now select all faces and unwrap them using U Unwrap > Follow Active Quads, should yield an even quad orthogonal grid.

Afterwards if you wish to make it fit inside de [0,1] UV space then either scale it down until if vaguely fits inside the image area, turn on UV > Constrain to Image Bounds and scale it up manually; or use the operator UV > Pack Islands to automatically fit maintaining proportions.